2. What is a binary tree?
• Property 1: each node can have up to two
successor nodes.
3. • Property 2: a unique path exists from the
root to every other node
What is a binary tree? (cont.)
Not a valid binary tree!
4. Some terminology
• The successor nodes of a node are called its children
• The predecessor node of a node is called its parent
• The "beginning" node is called the root (has no parent)
• A node without children is called a leaf
5. Some terminology (cont’d)
• Nodes are organize in levels (indexed from 0).
• Level (or depth) of a node: number of edges in the path
from the root to that node.
• Height of a tree h: #levels = L
(Warning: some books define h
as #levels-1).
• Full tree: every node has exactly
two children and all the
leaves are on the same level.
not full!
6. What is the max #nodes
at some level l?
The max #nodes at level is
l 2l
0
2
1
2
2
2
3
2
where l=0,1,2, ...,L-1
7. What is the total #nodes N
of a full tree with height h?
0 1 1
1
0 1 1 1
1
0
2 2 ... 2 2 1
...
n
h h
n
n i x
x
i
N
x x x x
using the geometric series:
l=0 l=1 l=h-1
8. What is the height h
of a full tree with N nodes?
2 1
2 1
log( 1) (log )
h
h
N
N
h N O N
9. Why is h important?
• Tree operations (e.g., insert, delete, retrieve
etc.) are typically expressed in terms of h.
• So, h determines running time!
10. •What is the max height of a tree with
N nodes?
•What is the min height of a tree with
N nodes?
N (same as a linked list)
log(N+1)
11. How to search a binary tree?
(1) Start at the root
(2) Search the tree level
by level, until you find
the element you are
searching for or you reach
a leaf.
Is this better than searching a linked list?
No O(N)
12. Binary Search Trees (BSTs)
• Binary Search Tree Property:
The value stored at
a node is greater than
the value stored at its
left child and less than
the value stored at its
right child
13. In a BST, the value
stored at the root of
a subtree is greater
than any value in its
left subtree and less
than any value in its
right subtree!
Binary Search Trees (BSTs)
14. Where is the
smallest element?
Ans: leftmost element
Where is the largest
element?
Ans: rightmost element
Binary Search Trees (BSTs)
15. How to search a binary search tree?
(1) Start at the root
(2) Compare the value of
the item you are
searching for with
the value stored at
the root
(3) If the values are
equal, then item
found; otherwise, if it
is a leaf node, then
not found
16. How to search a binary search tree?
(4) If it is less than the value
stored at the root, then
search the left subtree
(5) If it is greater than the
value stored at the root,
then search the right
subtree
(6) Repeat steps 2-6 for the
root of the subtree chosen
in the previous step 4 or 5
17. How to search a binary search tree?
Is this better than searching
a linked list?
Yes !! ---> O(logN)
21. Function NumberOfNodes
• Recursive implementation
#nodes in a tree =
#nodes in left subtree + #nodes in right subtree + 1
• What is the size factor?
Number of nodes in the tree we are examining
• What is the base case?
The tree is empty
• What is the general case?
CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1
22. template<class ItemType>
int TreeType<ItemType>::NumberOfNodes() const
{
return CountNodes(root);
}
template<class ItemType>
int CountNodes(TreeNode<ItemType>* tree)
{
if (tree == NULL)
return 0;
else
return CountNodes(tree->left) + CountNodes(tree->right) + 1;
}
Function NumberOfNodes (cont.)
O(N)
Running Time?
24. Function RetrieveItem
• What is the size of the problem?
Number of nodes in the tree we are examining
• What is the base case(s)?
1) When the key is found
2) The tree is empty (key was not found)
• What is the general case?
Search in the left or right subtrees
25. template <class ItemType>
void TreeType<ItemType>:: RetrieveItem(ItemType& item, bool& found)
{
Retrieve(root, item, found);
}
template<class ItemType>
void Retrieve(TreeNode<ItemType>* tree, ItemType& item, bool& found)
{
if (tree == NULL) // base case 2
found = false;
else if(item < tree->info)
Retrieve(tree->left, item, found);
else if(item > tree->info)
Retrieve(tree->right, item, found);
else { // base case 1
item = tree->info;
found = true;
}
}
Function RetrieveItem (cont.)
O(h)
Running Time?
28. • What is the size of the problem?
Number of nodes in the tree we are examining
• What is the base case(s)?
The tree is empty
• What is the general case?
Choose the left or right subtree
Function InsertItem (cont.)
33. • Unbalanced trees are not desirable because
search time increases!
• Advanced tree structures, such as red-black
trees, guarantee balanced trees.
Does the order of inserting elements
into a tree matter? (cont’d)
34. Function DeleteItem
• First, find the item; then, delete it
• Binary search tree property must be
preserved!!
• We need to consider three different cases:
(1) Deleting a leaf
(2) Deleting a node with only one child
(3) Deleting a node with two children
38. • Find predecessor (i.e., rightmost node in the
left subtree)
• Replace the data of the node to be deleted
with predecessor's data
• Delete predecessor node
(3) Deleting a node with two
children (cont.)
39. • What is the size of the problem?
Number of nodes in the tree we are examining
• What is the base case(s)?
Key to be deleted was found
• What is the general case?
Choose the left or right subtree
Function DeleteItem (cont.)
45. Tree Traversals
There are mainly three ways to traverse a tree:
1) Inorder Traversal
2) Postorder Traversal
3) Preorder Traversal
46. 46
Inorder Traversal: A E H J M T Y
‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’ ‘Y’
tree
Visit left subtree first Visit right subtree last
Visit second
47. Inorder Traversal
• Visit the nodes in the left subtree, then visit
the root of the tree, then visit the nodes in
the right subtree
Inorder(tree)
If tree is not NULL
Inorder(Left(tree))
Visit Info(tree)
Inorder(Right(tree))
Warning: "visit" implies do something with the value at
the node (e.g., print, save, update etc.).
48. 48
Preorder Traversal: J E A H T M Y
‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’ ‘Y’
tree
Visit left subtree second Visit right subtree last
Visit first
49. Preorder Traversal
• Visit the root of the tree first, then visit the
nodes in the left subtree, then visit the nodes
in the right subtree
Preorder(tree)
If tree is not NULL
Visit Info(tree)
Preorder(Left(tree))
Preorder(Right(tree))
51. Postorder Traversal
• Visit the nodes in the left subtree first, then
visit the nodes in the right subtree, then visit
the root of the tree
Postorder(tree)
If tree is not NULL
Postorder(Left(tree))
Postorder(Right(tree))
Visit Info(tree)
53. Function PrintTree
• We use "inorder" to print out the node values.
• Keys will be printed out in sorted order.
• Hint: binary search could be used for sorting!
A D J M Q R T
60. ResetTree and GetNextItem
• User needs to specify the tree
traversal order.
• For efficiency, ResetTree stores in
a queue the results of the
specified tree traversal.
• Then, GetNextItem, dequeues the
node values from the queue.
void ResetTree(OrderType);
void GetNextItem(ItemType&,
OrderType, bool&);
61. Revise Tree Class Specification
enum OrderType {PRE_ORDER, IN_ORDER, POST_ORDER};
template<class ItemType>
class TreeType {
public:
// previous member functions
void PreOrder(TreeNode<ItemType>, QueType<ItemType>&)
void InOrder(TreeNode<ItemType>, QueType<ItemType>&)
void PostOrder(TreeNode<ItemType>, QueType<ItemType>&)
private:
TreeNode<ItemType>* root;
QueType<ItemType> preQue;
QueType<ItemType> inQue;
QueType<ItemType> postQue;
};
new private data
new member
functions